## Mathematical Tool of Quantum Mechanics

**Problem: **

Consider the set of all entities of the form where the entries are real numbers. Addition and scalar multiplication are defined as follows:

i.) Write down the null vector and inverse of .

ii.) Show that vectors of the form do not form a vector space.

*Solution: *

**i.)** Let be the null vector of , such that

Then as defined,

and by comparison,

If we are to solve these equations for the values of and , we could get

.

Therefore, the * null vector *is given as

.

Now, we let the inverse of be , such that following the definition of an inverse vector we have

Following the same process as before, if we compare we can obtain

Solving for and ,

.

Therefore, the **inverse vector*** *of is

**.**

**ii.)** The vector do not form a vector space because:

a.) it violates the *closure *under addition,

.

b.) it violates the closure under scalar multiplication,

.

c.) there is **NO **null vector

d. an additive inverse does not exist

.

#### By: Joshua J. Ordeniza, MS Physics Student, MSU-IIT